If d/dt is velocity, what is d/dx ?

Well, it is easy to understand if we regard time as a dimension, which it is according to relativity. Further, space and time are on equal footing, i.e., what you can say about time you can also say that about space, and vice versa. Pictorially we draw the 3-dimensional coordinate system and add one more (4th) dimension to include time. So as the time derivative d/dt tells us the rate of change of U w.r.t. time (t-axis), in the same way, d/dx tells us the rate of change of U w.r.t x-axis.
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